THE PRECESSING ORBIT OF ECE FLUID DYNAMICS. M. W. Evans and H. Eckardt

Transcription

1 THE PRECESSING ORBIT OF ECE FLUID DYNAMICS by M. W. Evans and H. Eckardt Civil List, AlAS and UPITEC ( \Vww.upitec.org,, W\VW.et3m.net) ABSTRACT The Binet equation of ECE fluid dynamics is used to derive a precessing planar orbit which can be compared with the general planar orbit derived in UFT328 by simultaneous solution ofthe hamiltonian and lagrangian ofece2 relativity. A three dimensional lagrangian analysis is used to show that the Corio lis accelerations of any planar orbit vanish as a result of a central potential. This is the origin of the Leibnitz equation of orbits. The methods are illustrated with the motion of stars in a whirlpool galaxy. Keywords: ECE2 fluid dynamics, precessing orbit, motion of stars in a whirlpool galaxy.

2 1. INTRODUCTION Recently in this series of papers and books { 1-12} it has been shown that ECE fluid dynamics applied to orbits results in a modified Binet equation in which the effect of the vacuum is analyzed by the presence of a well defined spin connection. In Section 2 of this paper a three dimensional lagrangian analysis is used to show that the vacuum, represented by the spin connection, results in a precessing planar orbit as observed experimentally. The experimental observations can be used to measure the spin connection. The essential method is to replace classical dynamics by fluid dynamics. The fluid is the vacuum or aether, and in ECE2 is defined by Cartan geometry. The precessing orbit derived analytically in this way is compared with the general precessing orbit derived numerically in UFT328 by simultaneous solution of the ECE2 lagrangian and hamiltonian. The methods are illustrated by an analysis ofthe motion of stars in a whirlpool galaxy. These can be outwards or inwards depending on basic definitions. Finally the force law of the precessing orbit is found from the modified Binet equation. This paper is a brief synopsis of detailed calculations found in the notes accompanying UFT365 on Note 365(1) defines the relevant three dimensional lagrangians for relativistic and non relativistic motion, note 365(2) analyzes the modified Binet equation, note 365(3) is the first step of an iterative procedure in which the force law of the precessing orbit is an inverse square law in the first approximation. The latter defines the precessing ellipse in terms of a spin connection ofece2 fluid dynamics. Note 365(4) shows that the Coriolis accelerations vanish for any planar orbit in conventional dynamics as the result of assuming a central potential of any kind. Note 365(5) illustrates the methods used in this paper by analyzing the motions of stars in a whirlpool galaxy. Finally Note 365(6) defines the force law for the precessing ellipse as the second step of the iterative procedure of

3 Note 365(3). 2. DERIVATION OF THE PRECESSING ORBIT Consider the Binet equation ofece2 fluid dynamics derived in UFT363: 1 ') _Vh'( L ") in which the spin connection represents the effect of the vacuum, aether, or spacetime and is defined by: -- The plane polar coordinate system ( r, e ) is used in Eq. ( 1.. ) for a mass m orbiting a mass M. Here Lis the constant angular momentum of the system and G is Newton's constant. In Eq. ( ~ ) the position of an element of the fluid spacetime is defined by: R,_ ~ ( < l t\ 8 ( *) )). ~ (!) In deriving Eq. ( 1.. ) it was assumed that: )R r0 o oe in the first approximation. More generally~ is a function ofr(t) and e ( ~. In the orbital theory of classical dynamics: _SL '0\ '( 0 _(s) In the first approximation assume that

4 in Eq. ( i ), i.e. that the force is a central inverse square law. Assume that the orbit is the precessing ellipse: where f(r) is a function to be determined. For very small precessions such as those in the solar system, the orbit is an ellipse\ ""'-- J. ( \ -\- {- ( S O _ (_ ~) 0 < J_ i&..._ ). - ( and standard analysis {I- 12} s:~ ot_\ ----=-") cl ) L (~:~~pi::)~ght~t;~o:~o~~9))* J(+troy(3({)e~ ~} i.e. l/j J_ - ( \O) dj)? -(l\ This is the equation: -(\~ where:

5 Its solution is: -... So: (o.5 8 ~---~-o') ( \-t -SL'., ;') It follows that: I - (1~. -(1~ \-\ o\( Therefore the precessing ellipse is: \ - -\ < J_ Q.E.D. In Section 3 this analytical result is compared with the numerical result ofuft328, obtained by simultaneous solution of the ECE2 hamiltonian and lagrangian. Experimentally: C'J bfr\& ~(\") ~J.,ci to high precision, where c is the vacuum speed of light. So the spin cmmection can be found experimentally for any planar orbit. As described in Note 365(4), a complete understanding of planar orbits in classical dynamics requires use of the three dimensional hamiltonian:

6 where the velocity is defined by: - J_, - - eu The three dimensional Eul: Lag1 range~~ ~"'-' )! _ ( "J_)\ <:J - -- ) - Aj d( - gives the force law: (.)7:\ r ~ ~ ~ ~ t.- ~~ -- For a central potential (one that depends only on r): 'll( ~ ~'(( ~< -\- J_~!a ~ ~ ~<' -(l~ - d(" (" 08 ~< so the force law becomes: e. lh plane polar coordinates:

7 _. and it follows that: JQ Q _.,!<- \ V'r\.( -( - - d( f - ~l~.e -c)~) - _, (e:l) _J1f R -(:J_~ -'f" d( and (<a t-l~ 8)3_e 0. -~6) - - Eq. ( 3 0 ) shows that the Corio lis acceleration vanishes for any planar orbit if it is assumed that the force law is central. Eq. ( d, ~)is the Leibnitz orbital equation, and can be To illustrate the use of the Binet equation consider Note 365(5) which analyses the hyperbolic spiral orbit: It follows from the Binet equation that: and that: \ - (o (3J.) f (<') L - 3 ~( Yh{ 0 ) -L~) _(3~) The force ( j3 ) is an outward centrifugal force, and Eq. ( ''-t ) shows that there is no force of attraction. In this case:

8 L~--l(o - & ~< M ~<;,. ej _L T -::. --":11,.--- -, (" - ~ ) L which shows that the stars are moving inwards. There is a contradiction between Eqs. (.).) ) and ( ~ ). This contradiction is resolved by reversing the sign of the angular velocity: ~ _ ~L -) - (?>t) so Eq. ( ~.(, ) is changed to: - l - Y\--..( Eqs. ( :>~ ) and ( ~~ ) self consistently show that the stars are moving outwards from a large central mass M. However, if it is assumed that: together with Eq. ( ~ l ), it follows that: e The centrifugal force from Eqs. (.)3:> ) and ( f ~'\ ) is: \..? 0!> V\o\.. { 0 and is positive valued, meaning that it is an attractive force. Eqs. (.)'\)and ( '-\-\ ) show that the stars are moving inwards. The motion of the stars in a whirlpool galaxy is complicated, and must be determined experimentally. The above is meant to convey a simple model analysis. ~

9 Finally, as in note 365(5), the force law for the orbit ( \ l ) is found using the modified Binet equation (!... ). Computer.algebra is used to eliminate human error, and the result shows that the force law is not that of the incorrect Einstein theory. 3. FURTHER ANALYTICAL AND COMPUTATIONAL ANALYSIS Section by Dr. Horst Eckardt

10 The precessing orbit of ECE uid dynamics M. W. Evans, H. Eckardt Civil List, A.I.A.S. and UPITEC ( Further analytical and computational analysis An elliptic orbit without precession is known to be r 0 (θ) = α 1 + ɛ cos(θ). (42) We will inspect the eect of dierent kinds of precession. In the simplied precession model of ECE theory the elliptic conical section with a constant precession is described by where r 1 (θ) = α 1 + ɛ cos(xθ) = α 1 + ɛ cos((1 A)θ) (43) x = 1 A (44) and A is a small positive constant. The general form with a non-constant angular factor is: α r 2 (θ) = 1 + ɛ cos(f(θ) θ). (45) From (17) we have: f(θ) = Ω 1 01r. (46) This means that the spin connection Ω is θ-dependent and is not constant. the exact form of f(θ) could be determined numerically from the numerical solution of the relativistic orbital problem as in UFT328. This is however dicult to extract, therefore we make an analytical approach that gives a behaviour similar to Fig. 8 in UFT328. We use f(θ) = 1 + A 2 (sin ((1 + 3A)θ) 2). (47) emyrone@aol.com mail@horst-eckardt.de 1

11 The dierent models for the x factor (44) and (47) are graphed in Fig. 1. The function f(θ) oscillates around the constant value of 1 A. For the model calculations we chose A = 0.05, α = 1, ɛ = 0.3. (48) The corresponding orbits r 0, r 1, r 2 in dependence of θ are graphed in Fig. 2. It can be seen that the maxima and minima (aphelion and perihelion) are mostly identical for both types of precession, but the complete orbits are dierent. This can also be seen from the polar orbit plots, Fig. 3 and Fig. 4 where the orbits r 1 and r 2 are compared with the non-precessiong orbit r 0. The variable factor f(θ) makes the orbit look more irregular, compared to the orbit with constant precession factor. The three types of orbit approaches r 0, r 1 and r 2 have been used in the Binet equation (1). The Newtonian orbit r 0 leads to L2 F 0 (r) = αmr0 2 (49) which is the Newtonian force law, rewritten with constants L and α. When inserting the conical section orbit (43) of x theory, the result is F 1 (r) = L2 (( A 2 2 A + 1 ) r 1 + ( 2 A A 2) α ) α m r 1 3. (50) This is obviously a combination of a 1/r 2 and 1/r 3 force as is known from earlier investigations. Using the more general form (47) leads to a highly complicated expression F 2 (r) which nevertheless can be plotted. The three forms of the Binet force equation are graphed in Fig. 5. The results resemble the orbits (Fig. 2), but there is a drop of Force for the case F 2 which may be induced by the analytical form (47). 2

12 Figure 1: Models for the orbital precession factor x. Figure 2: Orbits r 0, r 1, r 2 in dependence of θ. 3

13 Figure 3: Polar plot of orbits r 0, r 1. Figure 4: Polar plot of orbits r 0, r 2. 4

14 Figure 5: Radial force from the Binet equation for r 0, r 1, r 2. 5

15 ACKNOWLEDGMENTS The British Government is thanked for a Civil List Pension, and the staff of AlAS and others for many interesting discussions. Dave Burleigh, CEO of Annexa, is thanked for voluntary posting, site maintenance, and maintenance of feedback software. Alex Hill is thanked for translation and broadcasting, and Robert Cheshire for broadcasting. REFERENCES { 1} M.W. Evans, H. Eckardt, D. W. Lindstrom and S. J. Crothers, "The Principles ofece" (open source on combined sites, \\<WW.aias.us and hardback Epubli Berlin. softback New Generation London, 2016, Spanish translation by Alex Hill). {2} M.W. Evans, H. Eckardt, D. W. Lindstrom and S. J. Crothers, "ECE2: The Second Paradigm Shift" (open source combined sites, Epubli Berlin, in prep. 2017, translated by Alex Hill). {3} M.W. Evans, S. J. Crothers, H. Eckardt and K. Pendergast, "Criticisms ofthe Einstein Field Equation" (UFT30 1 on combined sites, Cambridge International 201 0). { 4} L. Felker, "The Evans Equations of Unified Field Theory" (UFT302 on combined sites, Abramis 2007, translated by Alex Hill). {5} M.W. Evans, "Collected Scientometrics" (UFT307 on combined sites, New Generation 2015). {6} M. W. Evans, H. Eckardt and D. W. Lindstrom, "Generally Covariant Unified Field", (open source combined sites, Abramis 2005 to 2011 in seven volumes). {7} H. Eckardt, "The ECE Engineering Model" (UFT303 on combined sites). {8} M.W. Evans, Ed., J. Found. Phys. Chern., (Cambridge Intemational2011). {9} M.W. Evans and L. B. Crowell, "Classical and Quantum Electrodynamics and the B(3) Field" (Open source Omnia Opera section ofwww.aias.us, and World Scientific 2001).

16 { 10} M. W. Evans and S. Kielich (Eds.), "Modem Nonlinear Optics" (Wiley Interscience, New York, 1992, 1993, 1997 and 2001), in two editions and six volumes. { 11} M. W. Evans and J.-P. Vigier, "The Enigmatic Photon" (Kluwer 1994 to 2002 and open source, Omnia Opera, in five volume hardback and five volumes soft back. {12} M.W. Evans and A. A. Hasanein, "The Photomagneton in Quantum Field Theory" (World Scientific, 1994).